$g(t) = (e^{2t}, t\sin(t))$ What is the velocity of $g(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(2e^{2t}, \sin(t) + t\cos(t))$ (Choice B) B $\sqrt{4e^{4t} + t^2\cos^2(t)}$ (Choice C) C $(2e^{2t}, -t\sin(t))$ (Choice D) D $(2e^{2t}, t\cos(t))$
Solution: The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t))$, then the velocity is: $f'(t) = (a'(t), b'(t))$ Our position function here is $g(t)$. $g'(t) = (2e^{2t}, \sin(t) + t\cos(t))$ Therefore, the velocity of $g(t)$ is $(2e^{2t}, \sin(t) + t\cos(t))$.